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In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any non identity element of the subgroup lies inside the subgroup. This property is useful in understanding group structures, normalizers, and Galois theory. Centrally closed subgroups are related to commutators and play a role in analyzing group symmetries and field extensions.
Some facts about centrally closed subgroups:
- Every malnormal subgroup is centrally closed.
- Every Frobenius kernel is centrally closed.
- SA subgroups are precisely the centrally closed Abelian subgroups.
- The trivial subgroup and the whole group are centrally closed
References
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